A counterexample to dispersive estimates for Schr\"odinger operators in higher dimensions

In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^{\alpha }$, $\alpha <\frac{n-3}{2}$, for which the solutions to the linear Schr"odinger equation in ${\mathbb{R}}^n$, $$-i{\mathrm{\partial }}_{t}u=-\mathrm{\Delta }u+Vu,u(0)=f,$$ do not obey the usual $L^1\to L^{\mathrm{\infty }}$ dispersive estimate. This contrasts with known results in dimensions $n \leq 3$,whereapointwisedecayconditionon$V$ is generally sufficient to imply dispersive bounds.